\documentclass[10pt,a4paper]{article} 

\usepackage{ctex} % 中文支持
\usepackage[top=2.5cm, bottom=2.5cm, left=2.5cm, right=2.5cm]{geometry} % 页边距
\usepackage{amsmath, amssymb} % 数学公式与符号
\usepackage{graphicx, color, url}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm} % 标题上移

\title{《基础复分析》第11章椭圆函数 - 习题}
\author{CGZ ET AL}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle 

%## 《基础复分析》习题十一

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 1

证明任意具有周期 $\omega_1, \omega_2$ 的偶椭圆函数可以表示为
$$
C \prod_{k=1}^{n} \frac{\mathcal{P}(z) - \mathcal{P}(a_k)}{\mathcal{P}(z) - \mathcal{P}(b_k)} \quad (C \text{ 为常数}),
$$
如果原点既不是零点, 也不是极点.
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 2

试证
$$
\mathcal{P}(z) - \mathcal{P}(u) = -\frac{\sigma(z-u)\sigma(z+u)}{\sigma(z)^2 \sigma(u)^2}.
$$
(提示: 证明右端是 $z$ 的周期函数, 比较 Laurent 展开式来确定乘法常数.)
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 3

试证
$$
\frac{\mathcal{P}'(z)}{\mathcal{P}(z) - \mathcal{P}(u)} = \zeta(z-u) + \zeta(z+u) - 2\zeta(z).
$$
(提示: 利用上一题并求对数导数.)
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 4

试证
$$
\mathcal{P}(z+u) = -\mathcal{P}(z) - \mathcal{P}(u) + \frac{1}{4} \left[ \frac{\mathcal{P}'(z) - \mathcal{P}'(u)}{\mathcal{P}(z) - \mathcal{P}(u)} \right]^2.
$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 5

试证
$$
\mathcal{P}(2z) = \frac{1}{4} \left[ \frac{\mathcal{P}''(z)}{\mathcal{P}'(z)} \right]^2 - 2\mathcal{P}(z).
$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 6

试证
$$
\mathcal{P}'(z) = -\frac{\sigma(2z)}{\sigma(z)^4}.
$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item % 7

试证
$$
\begin{vmatrix}
\mathcal{P}(z) & \mathcal{P}'(z) & 1 \\
\mathcal{P}(u) & \mathcal{P}'(u) & 1 \\
\mathcal{P}(u+z) & -\mathcal{P}'(u+z) & 1
\end{vmatrix} = 0.
$$



\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

